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An introduction to mathematical cryptography 1st Edition by Jeffrey Hoffstein, Jill Pipher, Joseph Silverman 0387779930 9780387779935

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ISBN 10: 0387779930 

ISBN 13: 9780387779935

Author: Jeffrey Hoffstein; Jill Pipher; Joseph H. Silverman

ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics – quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.

Table of contents:

  1. An Introduction to Cryptography
  2. Simple substitution ciphers
  3. Divisibility and greatest common divisors
  4. Modular arithmetic
  5. Prime numbers, unique factorization, and finite fields
  6. Powers and primitive roots in finite fields
  7. Cryptography before the computer age
  8. Symmetric and asymmetric ciphers
  9. Exercises
  10. Discrete Logarithms and Diffie–Hellman
  11. The birth of public key cryptography
  12. The discrete logarithm problem
  13. Diffie–Hellman key exchange
  14. The ElGamal public key cryptosystem
  15. An overview of the theory of groups
  16. How hard is the discrete logarithm problem?
  17. A collision algorithm for the DLP
  18. The Chinese remainder theorem
  19. The Pohlig–Hellman algorithm
  20. Rings, quotients, polynomials, and finite fields
  21. Exercises
  22. Integer Factorization and RSA
  23. Euler’s formula and roots modulo pq
  24. The RSA public key cryptosystem
  25. Implementation and security issues
  26. Primality testing
  27. Pollard’s bold0mu mumu ppunitspppp-1 factorization algorithm
  28. Factorization via difference of squares
  29. Smooth numbers and sieves
  30. The index calculus and discrete logarithms
  31. Quadratic residues and quadratic reciprocity
  32. Probabilistic encryption
  33. Exercises
  34. Combinatorics, Probability, and Information Theory
  35. Basic principles of counting
  36. The Vigenère cipher
  37. Probability theory
  38. Collision algorithms and meet-in-the-middle attacks
  39. Pollard’s bold0mu mumu units method
  40. Information theory
  41. Complexity Theory and P versus NP
  42. Exercises
  43. Elliptic Curves and Cryptography
  44. Elliptic curves
  45. Elliptic curves over finite fields
  46. The elliptic curve discrete logarithm problem
  47. Elliptic curve cryptography
  48. The evolution of public key cryptography
  49. Lenstra’s elliptic curve factorization algorithm
  50. Elliptic curves over F2k and over F2k
  51. Bilinear pairings on elliptic curves
  52. The Weil pairing over fields of prime power order
  53. Applications of the Weil pairing
  54. Exercises
  55. Lattices and Cryptography
  56. A congruential public key cryptosystem
  57. Subset-sum problems and knapsack cryptosystems
  58. A brief review of vector spaces
  59. Lattices: Basic definitions and properties
  60. Short vectors in lattices
  61. Babai’s algorithm
  62. Cryptosystems based on hard lattice problems
  63. The GGH public key cryptosystem
  64. Convolution polynomial rings
  65. The NTRU public key cryptosystem
  66. NTRU as a lattice cryptosystem
  67. Lattice reduction algorithms
  68. Applications of LLL to cryptanalysis
  69. Exercises
  70. Digital Signatures
  71. What is a digital signature?
  72. RSA digital signatures
  73. ElGamal digital signatures and DSA
  74. GGH lattice-based digital signatures
  75. NTRU digital signatures
  76. Exercises
  77. Additional Topics in Cryptography
  78. Hash functions
  79. Random numbers and pseudorandom number generators
  80. Zero-knowledge proofs
  81. Secret sharing schemes
  82. Identification schemes
  83. Padding schemes and the random oracle model
  84. Building protocols from cryptographic primitives
  85. Hyperelliptic curve cryptography
  86. Quantum computing
  87. Modern symmetric cryptosystems: DES and AES

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